J F Ritt wrote this work and it was published in 1932 by the American Mathematical Society. It is an early look at what will be called differential field theory. In 1948 he gave the book a massive revision, and the second edition was published in 1950 by the American Mathematical Society. The second edition changes the terminology to match the developing nomenclature of this growing area of mathematics. This first edition is good, but I would reccomend that one interested in this area read the second edition. The second edition, in my opinion, is a great work. Yet let it be said that the field has moved on greatly since 1948. Another great work in this area is E. R. Kolchin, Differential Algebra and Algebraic Groups, 1973. There are, of course, numerous modern works in this field, but I consider Ritt's second edition and Kolchin to be the two historically significant ones.